A processing machine with redundant actuators includes multiple actuators for jointly positioning a controlled parameter. For example, the controlled parameter can be a position of a worktool along each axis of motion, and the joint motion of the redundant actuators position such a worktool. In a number of implementations, the position of the worktool is the algebraic sum of the position of at least a first actuator and a position of a second actuator of the redundant actuators. Thus, the machine is over-actuated, and degrees of freedom are available to optimize the movement of the worktool along a desired processing pattern. The worktool can be positioned by independent operations of the redundant actuators, and thus the task of positioning the worktool along the processing pattern can be separated between redundant actuators. One example of such a machine is a laser processing machine with redundant actuators, in which the worktool is a laser beam.
Some conventional methods, see, e.g., U.S. Pat. Nos. 5,452,275 and 7,710,060, use frequency separation techniques to assign the task of positioning the laser beam to the redundant actuators. For example, a low pass filter filters the processing pattern. The filtered signal becomes a reference trajectory for one actuator, while a difference between the processing pattern and the filtered signal becomes a reference trajectory for another actuator. In those methods, the redundant actuators jointly and supportively position the worktool.